U.S. authorities are pushing BNP Paribas SA to pay more than $10 billion to end a criminal probe into allegations the bank evaded U.S. sanctions, according to people familiar with the negotiations, a sign of increasing pressure the government is putting on large financial firms it suspects of wrongdoing.

A settlement of $10 billion or more, along with the guilty plea prosecutors are also seeking, would eclipse the punishment inflicted on other banks for similar conduct. To date, the most severe penalty on a bank suspected...